Majorana fermions, particles which are their own antiparticles, were originally envisioned by E. Majorana in 1937 in the context of particle physics (i.e., the physics of neutrinos). However, the current search for Majorana modes (Majoranas) is mostly taking place in condensed matter systems where Majorana quasi-particles appear in electronic systems as a result of fractionalization, and as emergent modes occupying non-local zero energy states. The non-locality of these modes provides the ability to exchange and manipulate fractionalized quasiparticles and leads to non-Abelian braiding statistics. Hence, in addition to being of paramount importance for fundamental physics, this property of the Majoranas places them at the heart of topological quantum computing schemes.
Majorana zero-energy modes/quasiparticles can appear quite naturally in 2D chiral p-wave superconductors where these quasiparticles, localized at the vortex cores, correspond to an equal superposition of a particle and a hole. A very simple model for Majorana zero-energy modes/quasiparticles is a one-dimensional (1D) Majorana quantum wire with localized Majorana zero-energy modes/quasiparticles at the ends. Both of the above cases involve spinless p-wave superconductors where the existence of Majorana zero-energy modes can be explicitly demonstrated by solving the corresponding mean field Bardeen-Cooper-Schrieffer (BCS) Hamiltonian. Recently, a way to engineer spinless p-wave superconductors has been suggested using a combination of strong spin-orbit coupling and superconducting proximity effect, thus opening the possibility of realizing Majorana quasiparticles in solid-state systems.